The trouble with palace intrigue like that is the big Heimdall-shaped obstacle in the way. Sure, he's not watching and listening all the time, but often enough that people actively worry about being heard. And given what he said in Thor, Loki seems to be the first person to figure out a way to conceal themselves from his senses; unless he can teach that to others, it won't help any. Especially since if Loki's prison is not extremely secure and always monitored, Odin is an idiot.

Plus let's not forget in the MCU only a handful of people know that Loki is actually a Frost Giant. Imagine what would happen if it was let out that one of their Asgardian princes was actually the son of the enemy? Sure you'll have a few open minded Asgardians but i think the majority of them would be in a state of uproar over that.

__________________- Stop making excuses for your naivety.

- I'm not a hero or a villain. But I am the destroyer of selfish hopes and dreams that don't benefit me at all.

- I hate it when people mess with my emotions and lead me on. I don't play games.

- Selfish people tend to forget that it is I that's the victim here, not them.